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Hyper-(Abelian-by-finite) groups with many subgroups of finite depth. (English) Zbl 1131.20024

A group has finite depth if its lower central series stabilizes after a finite number of steps. The authors prove that a finitely generated hyper-(Abelian-by-finite) group \(G\) is finite-by-nilpotent if and only if every of its infinite subsets contains two distinct elements \(x\) and \(y\) such that the subgroup generated by \(x\) and \(x^y\) has finite depth. They also prove a similar result with the condition that the subgroup generated by \(x\) and \(x^y\) is a min-\(n\)-by-Engel group.

MSC:

20F22 Other classes of groups defined by subgroup chains
20F19 Generalizations of solvable and nilpotent groups
20F14 Derived series, central series, and generalizations for groups
20F05 Generators, relations, and presentations of groups
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References:

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